LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A finite verifier induces an effect on its witness

Statement

For every verifier interface of Uniform verifier interface with a quantum witness, the compression $E_{x} = J_{x}^{*} U_{n}^{*} P_{x} U_{n} J_{x}$ is a positive contraction on $W_{n}$ . For every witness density operator $ρ$ , its terminal acceptance probability, summed over every unmeasured wire, equals $Tr (E_{x} ρ) \in [0, 1]$ .

Facts & Assumptions

Given: The finite input and witness registers and zero-initialized workspace, circuit unitary $U_{n}$ , accepting output projection $P_{x}$ , and witness embedding $J_{x}$ .

[F1]

Uniform verifier interface with a quantum witness: $J_{x}$ embeds $ψ$ as $∣ x ⟩ \otimes ψ \otimes ∣ 0^{a (n)} ⟩$ , $U_{n}$ is unitary, and $E_{x} = J_{x}^{*} U_{n}^{*} P_{x} U_{n} J_{x}$ .

[F2]

The partial trace is characterized by the trace pairing and is basis-independent: $Tr ({Tr}_{B} R T) = Tr (R (T \otimes I_{B}))$ for finite tensor factors.

[F3]

For $A \in M_{m \times n} (F)$ and $B \in M_{n \times m} (F)$ , $tr (A B) = tr (B A)$ : the finite matrix trace satisfies $Tr (A B) = Tr (B A)$ for compatible rectangular matrices.

[F4]

POVM outcome weights are probabilities: for a density operator and a finite POVM, its weights $Tr (ρ E_{a})$ are nonnegative and sum to one.

[F5]

Finite POVMs and their outcome law: a finite POVM has positive effects summing to identity, and its outcome weight on $ρ$ is $Tr (ρ E_{a})$ .

[F6]

Partial trace over a finite tensor factor: the partial trace over a finite tensor factor is defined by summing equal-index matrix entries in the ordered product basis.

[F7]

Density operator: a density operator is nonnegative and has trace one on a nonzero finite-dimensional space.

Proof

1.1givenF1algebra

Put $Q = U_{n} J_{x}$ . The input basis vector and clean zero state are unit vectors, so $J_{x}$ is an isometry; $U_{n}$ is unitary, hence $Q^{*} Q = I$ . Then $E_{x} = Q^{*} P_{x} Q$ and $I - E_{x} = Q^{*} (I - P_{x}) Q$ . For every $v \in W_{n}$ , $⟨ E_{x} v, v ⟩ = ⟨ P_{x} Q v, Q v ⟩ = ∥ P_{x} Q v ∥^{2} \geq 0$ , and $⟨ (I - E_{x}) v, v ⟩ = ∥ (I - P_{x}) Q v ∥^{2} \geq 0$ . Therefore $0 \leq E_{x} \leq I$ .

2.1step 1.1F4F5F7

For a density operator $ρ$ , [F7] gives the state hypotheses for the two-outcome POVM $(E_{x}, I - E_{x})$ defined in [F5]. Applying [F4] gives $Tr (E_{x} ρ) \in [0, 1]$ .

3.1givenF1F2F3F5F6F7algebra∎

The post-circuit state is $σ = U_{n} (J_{x} ρ J_{x}^{*}) U_{n}^{*}$ . By [F6] and the partial-trace identity [F2], the probability of the accepting projection on the named output register equals $Tr (P_{x} σ)$ , where $P_{x}$ is the accepting projection tensored with identity on all unmeasured wires; [F5] gives this projective measurement outcome law. Cyclicity [F3] and finite matrix multiplication give $Tr (P_{x} U_{n} J_{x} ρ J_{x}^{*} U_{n}^{*}) = Tr (J_{x}^{*} U_{n}^{*} P_{x} U_{n} J_{x} ρ) = Tr (E_{x} ρ)$ . The projection $P_{x}$ sums the probabilities over all unmeasured basis strings, so this is the stated terminal acceptance probability.

Depends on

Used by

Dependency tree · two levels

41 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John Watrous, Quantum Computational Complexity (standard reference, not scraped)
John Watrous, The Theory of Quantum Information (standard reference, not scraped)